A State Space Modeling Approach to Mediation Analysis

A State Space Modeling Approach

to Mediation Analysis

Fei Gu

McGill University

Kristopher J. Preacher

Vanderbilt University

Emilio Ferrer

University of California, Davis

Mediation is a causal process that evolves over time. Thus, a study of

mediation requires data collected throughout the process. However, most

applications of mediation analysis use cross-sectional rather than longitudi-

nal data. Another implicit assumption commonly made in longitudinal designs

for mediation analysis is that the same mediation process universally applies

to all members of the population under investigation. This assumption ignores

the important issue of ergodicity before aggregating the data across subjects.

We first argue that there exists a discrepancy between the concept of media-

tion and the research designs that are typically used to investigate it. Second,

based on the concept of ergodicity, we argue that a given mediation process

probably is not equally valid for all individuals in a population. Therefore, the

purpose of this article is to propose a two-faceted solution. The first facet of

the solution is that we advocate a single-subject time-series design that aligns

data collection with researchers’ conceptual understanding of mediation. The

second facet is to introduce a flexible statistical method—the state space

model—as an ideal technique to analyze single-subject time series data in

mediation studies. We provide an overview of the state space method and illus-

trative applications using both simulated and real time series data. Finally, we

discuss additional issues related to research design and modeling.

Keywords: mediation, state space model

1. Introduction

Mediation is a causal process that evolves over time. In the simplest case, the

causal variable (X) exerts an effect on the outcome variable (Y) partially or

Journal of Educational and Behavioral Statistics

2014, Vol. 39, No. 2, pp. 117–143

DOI: 10.3102/1076998614524823

# 2014 AERA. http://jebs.aera.net

117

completely through a mediator variable (M) over time. Clearly, time plays an

explicit role in the mediation process. In one of the earliest articles devoted spe-

cifically to mediation, Judd and Kenny (1981) emphasized the role of time, even

placing the words ‘‘process analysis’’ in the title of their article. More recently,

Schmitz (2006) provided an overview of the necessity of process analyses in the

context of learning and instruction. Ideally, the empirical study of mediation

requires (1) data collected throughout the process and (2) pertinent statistical

methods that can capture the dynamic mechanism underlying the causal process.

However, process-oriented methods that explicitly consider the role of time are

not common in educational and psychological research (Schmitz, 2006). In the

context of mediation, Cole and Maxwell (2003) discussed that most applications

of mediation analysis use cross-sectional rather than longitudinal data, and not all

methodological treatments acknowledge the necessity of considering time

(see Maxwell & Cole, 2007; Maxwell, Cole, & Mitchell, 2011).

Another implicit assumption commonly made in mediation analysis is that the

same mediation process universally applies to all members of the population

under investigation. This assumption is made in a handful of longitudinal models

recently developed within the structural equation modeling (SEM) framework

proposed to study mediation processes. Cheong, MacKinnon, and Khoo (2003)

used a parallel process latent growth curve model to investigate the effect of a

causal variable on the change in an outcome variable through change in a med-

iator variable. Gollob and Reichardt (1991) and Cole and Maxwell (2003)

implemented a cross-lagged panel model emphasizing longitudinal relations

between the absolute level of the causal variable and the outcome variable

through the mediator variable. Another variant of a longitudinal SEM approach

is the latent difference score model, in which differences between adjacent obser-

vations are treated as latent variables (e.g., Hamagami & McArdle, 2007;

MacKinnon, 2008; McArdle, 2001; McArdle & Nesselroade, 1994; Selig &

Preacher, 2009). In the traditional regression framework, Judd, Kenny, and

McClelland (2001) advocated the use of within-subject designs (in contrast to

between-subjects designs) to assess mediation and moderation, where individu-

als are put into both treatments. In all models mentioned previously, parameters

are estimated by pooling information across subjects. Although the longitudinal

designs in which these models are applied take the role of time into consider-

ation, they are still limited in that none of them considers the important issue

of ergodicity before aggregating the data across subjects (ergodicity is described

later). As we will show shortly, the conclusions drawn from pooling across sub-

jects may not be informative about how single subjects behave (e.g., Ferrer &

Widaman, 2008; Molenaar, 2004).

In this article, we contend that (a) there is a discrepancy between the concept

of mediation and the research designs that are typically undertaken to investigate

it in practice and (b) based on the concept of ergodicity, there is not necessarily a

universal mediation process that is equally valid for everyone in the population.

A State Space Modeling Approach to Mediation Analysis

118

Therefore, the purpose of this article is to propose a two-faceted solution to the

persistent problem of using cross-sectional designs in mediation analysis. The

first facet of the solution is to encourage researchers to rethink how they

approach the research design for mediation studies; that is, we advocate moving

from the traditional cross-sectional design to a single-subject time-series design

that aligns data collection with researchers’ conceptual understanding of media-

tion. The second facet is to introduce a flexible statistical method—the state

space model (SSM)—as an ideal technique to analyze single-subject1 time-

series data in mediation studies. In Section 2, we elaborate on the theoretical

foundation of the single-subject time-series design for mediation analysis. In

Section 3, we provide an overview of state space methods and two illustrative

applications using simulated and empirical data sets. In Section 4, we discuss

additional issues related to research design and modeling.

2. Foundation of the Single-Subject Time-Series Design

In a cross-sectional design, the focus is on the analysis of interindividual

variation—differences among different units at a single point in time. According

to a survey in 2005 of the five American Psychological Association journals pub-

lishing the most articles studying mediation, Maxwell and Cole (2007) reported

that more than half of the mediation studies were based on cross-sectional data.

However, according to the concept of mediation as well as the basic requirements

for causal inference, mediation must involve at least two relations that unfold

over time. Specifically, the effect of a causal variable (X) is first exerted on the

mediator variable (M, i.e., X!M), and then, this effect is carried over to the out-

come variable (Y, i.e., M! Y). Thus, it is immediately clear that a certain amount

of time must elapse for the effect of X to reach Y. Therefore, a requirement for

any mediation analysis is the consideration of the role of time, that is, the neces-

sity of the analysis of intraindividual variation—changes in the same unit over

time. By comparing the concept of mediation and how mediation analysis was

conducted in the literature, it is evident that there exists a large discrepancy

between how mediation is theoretically conceptualized and how it has actually

been modeled in the past. This discrepancy raises an important validity issue con-

cerning the equivalence between the analysis of interindividual variation and the

analogous analysis of intraindividual variation.

Cole and Maxwell (2003) argued and demonstrated that very restrictive con-

ditions are required to ensure accurate results from mediation analysis based on

cross-sectional data. In reality, such restrictive conditions almost never occur,

and bias in cross-sectional analyses of longitudinal mediation has been amply

demonstrated (Maxwell & Cole, 2007; Maxwell et al., 2011). The divide between

interindividual variation and intraindividual variation not only exists in media-

tion analysis but also appears in areas such as test theory, factor analysis, and

developmental psychology (Molenaar, 2004, 2008a, 2008b).

Gu et al.

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Given the existence of this divide in various areas, an important question

becomes whether this division between orientations is justified. Unfortunately,

as a direct consequence of the classical ergodic theorems, the answer to the ques-

tion is ‘‘no.’’ In fact, equivalence between the analysis of inter- and intraindivi-

dual variation is established only for ergodic processes (Molenaar, 2004). It is

worth noting that, outside the context of mediation and in the broader sense of

studying behavior, the inter-/intraindividual debate can be traced back to an older

distinction between nomothetic lawfulness, emphasizing generality in the popu-

lation, and idiographic characterization, emphasizing the uniqueness of the indi-

vidual (e.g., Allport, 1937; Lamiell, 1981, 1988; Molenaar, 2004; Rosenzweig,

1958; van Kampen, 2000; Zevon & Tellegen, 1982). When individuals differ

qualitatively rather than quantitatively, ‘‘Qualitative differences mistaken for

quantitative differences can seriously distort relationships and are a prescription

for diluted nomothetic relationships’’ (Nesselroade, Gerstorf, Hardy, & Ram,

2007, p. 219). The following subsection gives a brief, heuristic description of the

concept of ergodicity as the foundation of the single-subject time-series design.

2.1. Ergodicity

From the perspective of dynamical systems, a process is said to be ergodic if

the average of a single trajectory over time (structure of intraindividual variation)

is equal to the average of the ensemble of trajectories at a single point in time

(structure of interindividual variation by pooling across subjects). In order to

understand the consequences of ergodicity in psychology, the development of

human behavior over time is conceived of as a unique high-dimensional space

that contains dynamic processes and all the relevant information about the sub-

ject (cf. Molenaar, 1994, 2004, 2008a, 2008b; Molenaar & Ram, 2009, 2010;

Sinclair & Molenaar, 2008). For a particular individual, a finite sample of the

dynamic process over consecutive time points (usually evenly spaced) constitu-

tes a trajectory in his or her behavior space. This trajectory carries information

about intraindividual variation. Correspondingly, a finite sample of the same

behavior space from a group of individuals at a single point in time represents

an ensemble of trajectories, carrying information about interindividual variation

from this group of individuals. Hence, the question of whether the divide

between orientations is justified becomes a question of whether the developmen-

tal trajectory of human behavior is ergodic. This, in turn, reduces to the question

of whether stationarity and homogeneity hold for a Gaussian process. Therefore,

examining the stationarity and homogeneity criteria for a given process are

essential empirical steps.

For a Gaussian process, two criteria are required to be met simultaneously for

a process to be considered ergodic: stationarity and homogeneity. In terms of the

first criterion, a stationary process refers to a stochastic process whose joint prob-

ability distribution is time-invariant. For a Gaussian process, stationarity2


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requires that the first two moments of the process are time-invariant. That is, the

mean function of the time series is a constant, and the covariance function of the

time series depends only on relative time differences (i.e., ‘‘lag’’). Nonstationary

processes, however, are the norm in psychology, for example, learning and

developmental trajectories.

Regarding the second criterion, homogeneity refers to the situation in which each

member of a population obeys the same dynamic law and follows the same statistical

model, constituting exchangeable replications of each other, much as molecules of a

homogeneous gas. The reality, however, is quite the opposite. In fact, heterogeneity

is a general characteristic of human populations. Furthermore, besides the widely

recognized genetic and environmental effects that cause heterogeneity, Molenaar,

Boomsma, and Dolan (1993) argued that there exists a third source of developmental

differences: self-organization of nonlinear epigenetic processes.

Since nonstationarity, heterogeneity, or both are thought to be the rule rather

than the exception in most psychological processes, such processes are then

nonergodic, which means that the structure of interindividual variation is not

equivalent to the structure of intraindividual variation. Therefore, we conclude

that there is no equivalence between measurement orientations in the majority

of cases. This implies that there are not necessary lawful relationships between

the analysis of inter- and intraindividual variation. Thus, in cases concerning pro-

cesses that unfold over time, statistical analyses should focus on intraindividual

variation, with greater emphasis on single-subject time-series designs. This con-

clusion, therefore, also applies to mediation analysis.

Discussions of the implications and consequences of ergodicity started to

appear in many areas of psychological research about two decades ago (e.g.,

Molenaar, 1994; Molenaar, 2004, 2008a, 2008b; Molenaar & Campbell, 2009;

Molenaar & Ram, 2009, 2010; Molenaar, Sinclair, Rovine, Ram, & Corneal,

2009; Nesselroade & Molenaar, 1999; Sinclair & Molenaar, 2008, and most

recently Hamaker, 2012). However, there is virtually no mention of ergodicity

in the mediation literature (but see Roe, 2012).

3. Overview of State Space Methods

After establishing the theoretical foundation of the time-series design, process-

oriented methods are required to analyze the time series data. In this section, we

discuss the second facet of our proposed solution, that is, an overview of SSM. This

is, we believe, the first effort to utilize SSM to investigate mediation. As the first

application of SSM in the context of mediation, we provide some essential basics

of the most straightforward and frequently discussed variant of SSM in the time-

series and econometrics literature, that is, the linear Gaussian SSM. Due to space

limitations, our introduction is brief. For more comprehensive treatments of the

state space methodology in general, we refer readers to Commandeur and

Koopman (2007), Harvey (1989), and Durbin and Koopman (2001).

Gu et al.

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3.1 History of State Space Modeling in Psychology and Its Application to

Mediation Analysis

State space methods have their origin in control theory, beginning with the

groundbreaking article by Kalman (1960). Applications in astronautics were ini-

tially developed (and are still used) for accurately tracking the position and velo-

city of moving objects such as aircraft, missiles, and rockets. Shortly after its

application in engineering, SSM also found application in time series analysis and

econometrics (e.g., Aoki, 1987; Harvey, 1989). More recently, quantitative social

and behavioral scientists have begun applying SSM because of its statistical flex-

ibility for evaluating both the measurement properties and the lead–lag relation-

ships among latent variables in psychological processes. Analytic similarities

and differences between the currently dominant SEM and the relatively newly

emerging SSM in the psychology literature are discussed by several authors. Spe-

cifically, MacCallum and Ashby (1986) noted that SSM is a special case of SEM,

while Otter (1986) showed the reverse. Chow, Ho, Hamaker, and Dolan (2010)

reconciled the two approaches and provided a more detailed discussion of the rela-

tive strengths and weaknesses of both approaches vis-a-vis their use in representing

intraindividual dynamics and interindividual differences.

Another line of research involving SSM in psychology is built upon the state

space representation of the dynamic factor model (DFM; Molenaar, 1985).

Dynamic factor analysis was proposed to combine P-technique factor analysis

(Cattell, 1963; Cattell, Cattell, & Rhymer, 1947) and time series analysis (Browne

& Nesselroade, 2005; Molenaar, 1985; Molenaar, de Gooijer, & Schmitz, 1992;

Nesselroade, McArdle, Aggen, & Meyers, 2002). Some recent work devoted to

methodological discussions and substantive applications of DFM can be found

in the psychology literature (e.g., Chow, Nesselroade, Shifren, & McArdle,

2004; Ferrer & Nesselroade, 2003; Hershberger, Corneal, & Molenaar, 1994;

Nesselroade & Molenaar, 1999; Sbarra & Ferrer, 2006; Shifren, Hooker, Wood,

& Nesselroade, 1997; Wood & Brown, 1994; Zhang & Browne, 2006). Given the

strong similarity between SSM and DFM, the two terms frequently are used inter-

changeably, and DFMs often are expressed in state space form to exploit better

parameter estimation properties (Hamaker, Dolan, & Molenaar, 2005; Ho, Ombao,

& Shumway, 2005; Ho, Shumway, & Ombao, 2006; Molenaar, 1994; Song & Fer-

rer, 2009, 2012; Zhang, Hamaker, & Nesselroade, 2008).

Although SSM models have been used in many areas of research, their appli-

cation in questions about mediation is not available in the literature. Statistically,

probably all existing longitudinal models for studying mediation can be repre-

sented in their state space form, and identical results can be obtained. However,

we do not pursue this direction because of the consequences of the ergodic the-

orems stated before. Our proposed approach, instead, is based on the specifica-

tion of an SSM for analyzing time series data from a single subject (especially

when the number of measurement occasions is large).


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In practice, modeling the mediation process as an SSM has important benefits.

First, it allows the researcher to investigate time-related sequences among vari-

ables (i.e., predictor ! mediator ! outcome), as the process represented by

these variables unfolds over time. Although some longitudinal structural equa-

tion models also allow such investigation, the state space approach provides a

better and more thorough depiction because some mediation processes need a

longer time to unfold. Second, SSM can accommodate complex specifications

such as measurement structures and second-order factors.3 Third, state space

analyses at the individual level provide a theoretically sound, bottom-up

approach to create homogeneous subpopulations. If a hypothetical model fits

separately the time series data from several subjects, a homogeneous subpopula-

tion can be created from the analyses at the individual level, and generalized con-

clusions can be drawn for this subpopulation. The bottom-up approach, however,

can be labor-intensive. More discussion of multiple-subject time series is given

in subsequent sections.

3.2. The Linear Gaussian SSM

Currently, there is no standard notation in the literature for SSM, and different

authors have different preferences. Based on the similarity between SEM and

SSM, we choose to use LISCOMP notation to present the formulation of SSM.

The benefit of using LISCOMP notation is that each matrix has gained a standard

interpretation in the literature, thus providing a convenient and familiar notation.

Let yt be a p-variate vector representing p manifest variables, Zt a q-variate vector

representing q-latent variables (p � q � 1), and t ¼ 1, 2, . . . , T denotes the time

point for the corresponding vector. The general linear Gaussian SSM contains a

measurement equation

yt ¼ tt þ LtZt þ et; et � MVNð0;�tÞ;

and a transition equation,

Zt ¼ at þ BtZt�1 þ zt; zt � MVNð0;CtÞ;

where tt is a p� 1 vector for intercepts, Lt is a p� q loading matrix, et is a p� 1

vector for measurement errors (also known as innovations in the time series lit-

erature), �t is a p � p diagonal covariance matrix, at is a q � 1 vector for means,

Bt is a q � q transition matrix, zt is a q � 1 vector for residuals, and Ct is a q � q

covariance matrix. The measurement errors and residuals are assumed to be seri-

ally independent and independent of each other at all time points. We denote as yt

the vertical vector that collects all parameters in tt, Lt, �t, at, Bt, and Ct, and the

subscript t means that yt is time varying, thus resulting in the time-varying SSM.

Applications of the time-varying SSM can be found in recent articles by Mole-

naar, Sinclair, Rovine, Ram, and Corneal (2009), Sinclair and Molenaar

(2008), and Chow, Zu, Shifren, and Zhang (2011).

Gu et al.

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Here we consider only the time-invariant SSM, with the understanding that

the model could be extended to include time-varying parameters. The measure-

ment and transition equations are thus simplified to

yt ¼ tþ LZt þ et; et � MVNð0;�ÞZt ¼ aþ BZt�1 þ zt; zt � MVNð0;CÞ;

in which the parameter vector, y, is time-invariant.

3.3. Parameter Estimation and the Kalman Filter

Unknown parameters of the linear Gaussian SSM are estimated via a recursive

algorithm, called the Kalman filter (KF). The KF4 algorithm is initialized with the

latent variable, Z0|0, and the associated covariance matrix, P0|0, and proceeds with

the prediction and filtering steps iteratively at each time point. When t¼ 1, the pre-

diction step gives the predicted latent variable and its covariance matrix, that is,

Z1j0 ¼ aþ BZ0j0

P1j0 ¼ BP0j0B0 þC:

As a byproduct, the one-step-ahead prediction error and its associated covariance

matrix are obtained, that is,

e1 ¼ y1 � y1j0 ¼ y1 � ðtþ LZ1j0ÞD1 ¼ LP1j0L

0 þ�:

Then, the filtering step uses the observed value at t ¼ 1 to update the predicted

values, givingK1 ¼ P1j0L

0D�11 ðKalman gain matrixÞ

Z1j1 ¼ Z1j0 þ K1e1 ¼ Z1j0 þ P1j0L0D�1

1 e1

P1j1 ¼ P1j0 � K1D1K01 ¼ P1j0 � P1j0L0D�1

1 LP1j0:

When t ¼ 2, Z1|1 and P1|1 are used in the prediction step, followed by the filtering

step to calculate Z2|2 and P2|2, and so on. In sum, for t¼ 1, 2, . . . , T, the recursive

KF algorithm can be written as

Ztjt�1 ¼ aþ BZt�1jt�1

Ptjt�1 ¼ BPt�1jt�1B0 þC

et ¼ yt � ytjt�1 ¼ yt � ðtþ LZtjt�1ÞDt ¼ LPtjt�1L

0 þ�

Kt ¼ Ptjt�1L0D�1

t

Ztjt ¼ Ztjt�1 þ Ktet ¼ Ztjt�1 þ Ptjt�1L0D�1

t et

Ptjt ¼ Ptjt�1 � KtDtK0t ¼ Ptjt�1 � Ptjt�1L

0D�1t LPtjt�1:

Inserting et and Dt at each time point into the log-density function of the multi-

variate normal distribution and summing all log-density functions, the prediction

error decomposition (PED; Schweppe, 1965) function is obtained:


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PED ¼ 1

2

XT

t¼1

�p logð2�Þ � log Dtj j � e0tD�1t et

� �:

Giving certain starting values in y and maximizing the PED function with respect

to y provides the parameter estimates.

3.4. Testing Mediation: Bootstrapping the Time Series Data

As described in the previous sections, early definitions of direct and indirect

effects in mediation analysis were based on cross-sectional designs (Baron &

Kenny, 1986). These definitions were theoretically inaccurate because of the lack

of consideration of time. Modern definitions of the concepts advocate that the

causal variable, the mediator variable, and the outcome variable of a mediation

process should be obtained at different occasions (Collins, Graham, & Flaherty,

1998; Gollob & Reichardt, 1991). Figure 1 illustrates an example of the simplest

three-variate model in which time is taken into account. This figure illustrates the

concepts of direct and indirect effects in mediation analysis. In this model, a is

defined as the direct effect of Xt�1 on Mt, b is the direct effect of Mt on Ytþ1, and

c is the direct effect of Xt�1 on Ytþ1. The indirect effect of Xt�1 on Ytþ1 is defined

as the product of a and b, that is, ab. For more complicated models, the indirect

effect can be a product of more parameters linking several different occasions.

In order to evaluate the direct effects, several statistical tests are available, for

example, the Wald test and the likelihood ratio test. Testing the significance of

the indirect effects is also of great importance but more difficult, and the common

direct methods just mentioned are not appropriate because of the nonnormality of

the sampling distribution of the indirect effect. As an alternative, the use of boot-

strap confidence intervals (CIs) is recommended (Bollen & Stine, 1990; Hayes,

2009; MacKinnon, Lockwood, & Williams, 2004; Preacher & Hayes, 2004,

2008a, 2008b; Shrout & Bolger, 2002). The standard nonparametric bootstrap

involves two steps. In the first step, a resample of size N is drawn with replace-

ment from the original sample. In the second step, model parameters are esti-

mated from this resample. The two steps are replicated B times (where B is

large), so that the sampling distribution of the statistic of interest can be obtained.

FIGURE 1. Direct and indirect effects in the simplest mediation model.

Gu et al.

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At the .95 level, the 2.5th and 97.5th percentiles are chosen to construct the CI to

conduct the significance test of a single parameter, or of a product of parameters,

by examining whether the CI excludes 0 (indicating a significant effect). Boot-

strapping the time series data, however, poses yet another difficulty because of

the temporal dependence of the time series data (Zhang & Browne, 2006). In gen-

eral, the standard nonparametric bootstrap is not appropriate for time series data

because it destroys the inherent time dependency in the data. In this section, we

introduce two bootstrap methods appropriate for SSM, namely the parametric

bootstrap and the residual-based bootstrap. Besides these two methods, there are

other approaches appropriate for SSM (e.g., Zhang & Chow, 2010).

The parametric bootstrap is essentially a Monte Carlo simulation, in which

repeated bootstrap samples are simulated from a specified model, where the esti-

mates from the original sample are treated as parameters. The underlying

assumption is that the specified model is correct in the population. To generate

each random sample, a number of steps are followed:

1. Generate Z0 from MVN (0, 100 � Iq).

2. Set the iteration number t ¼ 1.

3. Generate zt from MVN (0, C).

4. Calculate Zt using Zt ¼ aþ BZt�1 þ zt.

5. Generate et from MVN (0, �).

6. Calculate yt using yt ¼ tþ LZt þ et.

7. Set t ¼ t þ 1 and return to Step 3.

8. Repeat Steps 3 to 6 until t > T þ 1,000.

9. Save the data from 1,001 to T þ 1,000.

Although not always necessary, the first 1,000 observations are typically

discarded as the burn-in period.

The residual-based bootstrap was first applied to linear Gaussian SSM by

Stoffer and Wall (1991) in assessing the precision of maximum likelihood esti-

mates, and it is considered a semiparametric approach. Similar to the parametric

bootstrap, population parameters are taken to be sample estimates in the

residual-based bootstrap, and the underlying assumption of a correctly specified

model is also made. On the other hand, random samples are drawn, with replace-

ment, from the standardized residuals as in the standard nonparametric bootstrap.

Specifically, the residual-based bootstrap procedure is based on the innovations

form of the KF:

et ¼ yt � t� LZtjt�1

Dt ¼ LPtjt�1L0 þ�

Kt ¼ Ptjt�1L0D�1

t

Ztþ1jt ¼ BZtjt�1 þ BKtet

yt ¼ tþ LZtjt�1 þ et:


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Then, the algorithm proceeds as follows:

1. Calculate standardized innovations using D�1=2t et, denoted ~et.

2. Draw, with replacement, a random sample from ~et to obtain ~e�t .

3. Construct a bootstrap sample by fixing the initial conditions of the KF and

iteratively using the following two equations:

Ztþ1jt ¼ BZtjt�1 þ BKtD1=2t ~e�t

yt ¼ tþ LZtjt�1 þ D1=2t ~e�t :

The idea behind the residual-based bootstrap is that the standardized residuals

are independent and identically distributed, and therefore exchangeable, after all

the dynamic and measurement relationships have been accounted for by the

model. This procedure, however, is not robust against model misspecification

(Stoffer & Wall, 1991, 2004; Zhang & Chow, 2010).

For the examples considered in this article, we set B ¼ 2,000 for both boot-

strap procedures to approximate the sampling distribution of the product. As a

general rule, large numbers are required to allow enough simulated cases in both

tails of the sampling distribution of the indirect effect so that the percentiles can

be accurately estimated for constructing a CI (Yung & Chan, 1999).

3.5. Illustration 1: A Simulated Lag-2 Example

In order to illustrate the parameter estimation in SSM and the two bootstrap

procedures just described, a three-variate, single-subject time series data set is

generated from a lag-2 model (depicted in Figure 2). Note that the figure repre-

sents a temporal slice of three time points from the entire process. Three equa-

tions are involved in this model:

FIGURE 2. A temporal slice of the lag-2 model.

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127

Yt ¼ aYt�1 þ bMt�1 þ cXt�1 þ gXt�2 þ eYt

Mt ¼ dMt�1 þ eXt�1 þ eMt

Xt ¼ fXt�1 þ eXt;

where a, d, and f are autoregressive parameters of the outcome variable, the med-

iator variable, and the causal variable, separately; b, c, and e are the lag-1 cross-

regressive parameters; g is the lag-2 cross-regressive parameter; and eYt, eMt, and

eXt are residuals in each equation. If g is equal to 0, it reduces to the lag-1 model,

which is a particular case of the lag-2 model. In addition, if c is also equal to 0, it

corresponds to one of the models in Cole and Maxwell (2003, model 5). As we

will show immediately, the lag-2 model can be expressed in state space form. In

principle, as long as we can write the model in state space form, the parameter

estimation and bootstrap procedures can be readily applied.

By defining the following measurement equation and the transition equation,

the state space form of the lag-2 model is obtained:

Yt

Mt

Xt

0B@

1CA ¼

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0B@

1CA

Yt

Mt

Xt

Yt�1

Mt�1

Xt�1

0BBBBBBBB@

1CCCCCCCCA; and � ¼

0 0 0

0 0 0

0 0 0

0B@

1CA

Yt

Mt

Xt

Yt�1

Mt�1

Xt�1

0BBBBBBBB@

1CCCCCCCCA¼

a b c 0 0 g

0 d e 0 0 0

0 0 f 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0BBBBBBBB@

1CCCCCCCCA

Yt�1

Mt�1

Xt�1

Yt�2

Mt�2

Xt�2

0BBBBBBBB@

1CCCCCCCCAþ

eYt

eMt

eXt

0

0

0

0BBBBBBBB@

1CCCCCCCCA;

and C ¼

cY 0 0 0 0 0

0 cM 0 0 0 0

0 0 cX 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0BBBBBBBB@

1CCCCCCCCA:

We simulated four time series data sets with lengths equal to 50, 100, 150, and

200. Compared with the typical lengths in the time series literature, the lengths

considered here are relatively short. Although most social scientists typically col-

lect at best only a handful of repeated measures, intensive longitudinal data are

highly desirable for SSM analyses. Some precedent examples have already

emerged in emotion studies using daily diary data (Chow, Hamaker, Fujita, &


128

Boker, 2009; Song & Ferrer, 2009, 2012), a cognitive study using daily cognitive

assessment data (Chow, Hamaker, & Allaire, 2009), and a neuropsychology

study using functional magnetic resonance imaging data (Ho et al., 2005).

We estimated the parameters using the KF algorithm with results shown in

Table 1. The estimated parameters in four different length conditions roughly

show that length of the time series is inversely related to the magnitude of para-

meter bias. A challenge, however, is that when the sampling frequency (i.e., the

time elapsed between consecutive measurement occasions) is fixed, longer

time series data are more expensive and difficult to collect. One strategy to

obtain longer time series data is to increase the sampling frequency. The issue

of length and sampling frequency will be discussed briefly in the last section.

The indirect effect from X(t�2) to Y(t) is the product of e and b. The two

bootstrap methods are used to construct 95% CIs for this product. For the four

simulated examples, the 95% CIs from both the parametric bootstrap and the

residual-based bootstrap are similar, and none of the CIs contains zero, indicat-

ing a significant lag-2 indirect effect from X(t�2) to Y(t). In addition, the CIs from

the parametric bootstrap are a bit larger than those from the residual-based

bootstrap when T ¼ 50 and 100; this difference almost disappears when

T ¼ 150 and 200.

TABLE 1

Parameter Estimates From a Lag-2 Model Fitted to Simulated Data

Parameter True Value T ¼ 50 T ¼ 100 T ¼ 150 T ¼ 200

a: autoreg of Y .5 .440 .460 .469 .432

b: Yt on Mt�1 .4 .295 .356 .351 .338

c: Yt on Xt�1 .4 .373 .342 .360 .386

d: autoreg of M .5 .414 .444 .463 .476

e: Mt on Xt�1 .4 .458 .462 .443 .400

f: autoreg of X .8 .820 .741 .778 .792

g: Yt on Xt�2 .3 .298 .327 .281 .309

cY .1 .110 .082 .091 .096

cM .4 .282 .318 .332 .349

cX .9 .753 .758 .815 .996

eb .16 .135 .164 .155 .135

resid parm resid parm resid parm resid parm

Number converged 2,000 2,000 2,000 2,000 2,000 2,000 2,000 2,000

Lower bound of eb .067 .055 .119 .110 .117 .115 .104 .105

Upper bound of eb .220 .245 .228 .237 .209 .208 .176 .178

Range of the 95% CI .153 .190 .109 .127 .092 .093 .072 .073

Note. CI ¼ confidence interval; resid ¼ residual-based bootstrap; parm ¼ parametric bootstrap; eb is

the indirect effect from Xt�2 to Yt.

Gu et al.

129

3.6. Illustration 2: An Empirical Study

In this section, we fit the lag-2 model displayed in Figure 2 to the time series of

two man–woman dyads. Each series contains self-reported daily stress and affect

for 91 days. These data are part of a larger study designed to examine dyadic inter-

actions (for more details see, e.g., Ferrer, Steele, & Hsieh, 2012; Ferrer &

Widaman, 2008). The variables used in these analyses are female perceived stress

(X), male positive affect specific to his relationship (M), and female negative affect

specific to her relationship (Y). Relationship-specific affect (RSA) was measured

using the RSA scale (Ferrer et al., 2012), 18 items intended to tap into the partici-

pants’ positive and negative emotional experiences specific to their relationships.

Examples of the positive items include ‘‘emotionally intimate,’’ ‘‘trusted,’’ and

‘‘loved.’’ Examples of the negative items include ‘‘sad,’’ ‘‘trapped,’’ and ‘‘discour-

aged.’’ The stress construct was measured using 5 items from the Positive and Neg-

ative Affect Schedule (Watson, Clark, & Tellegen, 1988) including ‘‘distress,’’

‘‘upset,’’ ‘‘scared,’’ ‘‘nervous,’’ and ‘‘afraid.’’ For all analyses, we created unit-

weighted composites for each person, using all the items in each of the scales.

The results of fitting the lag-2 models to these data are reported in Table 2. For

Dyad 1, the estimated values for parameters b and e are statistically significant,

indicating reliable evidence to support positive relations between the male

partner’s positive affect on a given day and his female partner’s negative affect

the following day, as well as between the female partner’s perceived stress on a

given day and the male partner’s positive affect the following day. However, the

estimated c and g parameters are not statistically significant. The nonsignificant

estimate of g may suggest a more parsimonious model that does not include the

lag-2 structure between the female partner’s stress and her subsequent negative

affect.

For Dyad 2, the estimated b parameter is not statistically significant (.011/.031

¼ .355), whereas the estimated e and g are both significant. It is not unreasonable

to expect that the nonsignificant estimate of b may result in a model with a

nonsignificant indirect effect linking the causal variable at t � 2 to the outcome

variable at t. For researchers interested in emotion, these preliminary results

provide motivation for further investigation of each dyad’s data separately. Par-

ticularly, the counterintuitive sign of the estimated b for both dyads may flag

some problems or uncertainties in the model, the data, or both. In reality, the

affective processes underlying different dyads may be qualitatively different

(e.g., including a feedback process).

For illustrative purposes, we also present the 95% CIs from the two bootstrap

methods for both dyads, but we acknowledge that the lag-2 model may not be

plausible for either dyad. The caveat is that these CIs are not readily interpretable

because neither bootstrap method is robust to model misspecification. Further

efforts are necessary to determine the best final model for each dyad and, in order

to make proper statistical inferences, the two bootstrap methods need to be


130

applied to each final model separately. This is beyond the scope of the current

illustration, but is deserving of separate study. In summary, the message from this

illustration for mediation researchers is clear; that is, mediation analyses should

be conducted for each subject (i.e., individual, dyad, or other unit of analysis)

separately to accommodate heterogeneity in the units.

4. Additional Issues

4.1. Causal Inference

Establishing causality is an important component of longitudinal research.

There are several recent treatments of causal inference in mediation analysis,

most inspired by Rubin’s causal model (or the potential outcomes framework;

see Albert, 2008; Ten Have & Joffe, 2010). In this framework, causality is

defined with reference to potential outcomes that might have been obtained

under different counterfactual conditions. Because it is not possible to observe

outcomes under all possible conditions, certain assumptions are commonly

invoked to permit causal inference, for example, the assumption that key paths

TABLE 2

Parameter Estimates From a Lag-2 Model Fitted to Two Dyads’ Time Series Over 91

Days

Parameter Dyad 1 Dyad 2

a: autoreg of Y (female negative affect relationship

specific)

.420 (.137) .267 (.121)

b: Yt on Mt�1 .220 (.043) .011 (.031)

c: Yt on Xt�1 .053 (.132) .473 (.214)

d: autoreg of M (male positive affect relationship specific) .707 (.049) .736 (.066)

e: Mt on Xt�1 .669 (.110) .792 (.205)

f: autoreg of X (stress) .927 (.040) .973 (.020)

g: Yt on Xt�2 �.042 (.099) .274 (.174)

cY: Var(pagf) .184 (.027) .113 (.017)

cM: Var(nasf) .385 (.057) .588 (.088)

cX: Var(pasf) .309 (.046) .049 (.007)

eb .147 .009

resid parm resid parm

Number converged 2,000 2,000 2,000 2,000

Lower bound of eb .100 .095 �.029 �.028

Upper bound of eb .225 .246 .076 .049

Range of the 95% CI .125 .151 .105 .077

Note. Standard errors are in parentheses. CI ¼ confidence interval; resid ¼ residual-based bootstrap;

parm ¼ parametric bootstrap; eb is the indirect effect from Xt�2 to Yt.

Gu et al.

131

composing an indirect effect are not confounded by omitted variables, and the

assumption that X does not moderate the effect of M on Y, among others (Imai,

Keele, & Tingley, 2010; Pearl, 2010, 2012; VanderWeele & Vansteelandt, 2009).

The single-subject design is not an attempt to establish causal relationships

per se. Instead, it provides a temporally plausible way to model and test a

hypothetical mediation process. That is, the parameters associated with the

mediator/mediators can be tested against a null hypothesis using a sampling

distribution (perhaps obtained by bootstrapping), so that the researcher can gain

more insights into the data and determine to what degree the data are consistent

with the hypothesized underlying process. Because it is not explicitly couched

in a potential outcomes framework, researchers should be cautious in making

causal inferences using the state space modeling approach to mediation analy-

sis. However, it should be noted that SSM has in its favor that key effects are

within-subject rather than between-subject, and measurements on key variables

are necessarily separated in time. These features provide a stronger basis for

causal inference than other methods that are designed for use with between-

subject and/or cross-sectional data. Extending the logic of the potential out-

comes framework to single-subject longitudinal designs is an interesting ave-

nue for future research.

4.2. Extension to the Multiple-Subject Time Series

Gathering time series data simultaneously on multiple persons (as in our

empirical study) is a common practice. In terms of modeling strategy, it can

be considered a straightforward extension to the model presented here. We call

this extension the multiple-subject time-series design. Since the introduction of

DFM (Molenaar, 1985), the single subject is always emphasized as the unit of

analysis. This emphasis may give researchers the impression that DFM is

restricted to analyzing the time series data of a single subject. This impression

is not incorrect for the standard application of DFM. As is demonstrated in the

modern literature, DFM and SSM can and should be extended to the multiple-

subject time-series design (e.g., Chow, Hamaker, & Allaire, 2009; Chow,

Hamaker, Fujita, & Boker, 2009; Chow, Zu et al., 2011; Hamaker et al., 2005;

Molenaar, 2010b; Nesselroade, 2010; Song & Ferrer, 2012).

The term panel model is often used to refer to models used to analyze time-

series data (of any length from short to long) collected from a group of subjects

(of any sample size). In mediation analysis, panel models under the SEM frame-

work usually take very short lengths (e.g., a handful of repeated measurements)

and large sample sizes (e.g., Cole & Maxwell, 2003).

Methodologically, multiple-subject modeling can be implemented easily by

fitting a qualitatively and quantitatively identical model to multiple subjects,

treating each individual as a group. By qualitatively identical, we mean that each

subject can be characterized by the same dynamic process implied by the


132

specified model; whereas by quantitatively identical, we mean that the para-

meters of the specified model are equated across different subjects. Then, it

is possible to test parameter invariance across subjects by means of likelihood

ratio tests. This procedure is akin to the standard multiple-group SEM analysis

of interindividual variation in searching for commonality across subjects

(nomothetic lawfulness). Following the new definition of parameter invariance

proposed by Nesselroade, Gerstorf, Hardy, and Ram (2007), we suggest that

parameter invariance tests can be better examined at an appropriate level of

abstraction. Specifically, parameters in the measurement equation can differ

to some arbitrary degree to recognize and isolate idiosyncrasy, while those in

the transition equation can be equated and tested for similarities that reflect

nomothetic lawfulness.

However, as described previously, some empirical examples suggest that

heterogeneity is the rule rather than the exception. In addition to our empirical

example, for instance, economists studying workers’ levels of satisfaction

encountered the problem that each individual anchors his or her scale at a dif-

ferent level (Winkelmann & Winkelmann, 1998). This renders interindividual

comparisons of responses meaningless in a cross-sectional study. Another

example, as discussed by Nesselroade (2010), concerns participants’ idiosyn-

cratic use of language. Specifically, one debriefed participant from Mitteness

and Nesselroade (1987) reported that she interpreted the term ‘‘anxious’’ to

mean ‘‘eager.’’

This empirical evidence further supports the general conclusion of heteroge-

neity across people. On the other hand, Kelderman and Molenaar (2007) pro-

vided counterintuitive evidence of the insensitivity of the standard factor

analysis of interindividual variation to the presence of extreme qualitative hetero-

geneity of the factor loadings in the population of subjects. This proven insensi-

tivity can have serious practical and ethical consequences, yielding individual

assessments and decisions that are biased to unknown degrees (Molenaar,

2008a, 2008b). Given these considerations, great caution should be used if homo-

geneity is to be assumed.

Another popular modeling strategy for multiple-subject time series data

that allows some degree of heterogeneity is the multilevel modeling frame-

work. Song and Ferrer (2012) recently proposed a random coefficient DFM

(which can be conceptualized as a multilevel SSM) to investigate both intra-

and interindividual variation. Specifically, they assumed that the parameters

in the transition matrix are drawn from some distribution, so that differences

in dynamics at the appropriate level of abstraction can be accommodated. A

further extension of their model is to allow the parameters in the loading

matrix to be random to capture heterogeneity in the between-subject factor

structure. Such an extension, however, is complex and difficult in terms of

model specification, and estimation may render a model infeasible and/or

indefensible in practice.

Gu et al.

133

4.3. Length and Sampling Frequency

Longer time series are desired for state space analysis and other time-series

techniques in general. Whereas increasing sampling frequency is a way to

obtain more data points for a given time span, the optimal balance of length and

sampling frequency is almost always context-dependent (Brose & Ram, 2012;

Sliwinski & Mogle, 2008). Researchers should consider carefully the relative

benefits of extending the length of a study versus making more observations

within a fixed length. A thorough discussion of the issue is beyond the scope

of this article; see Collins and Graham (2002), Nesselroade (1991), Nessel-

roade and Boker (1994), Nesselroade and Jones (1991), and Windle and Davies

(1999).

4.4. Use of Latent Variables in Mediation Analysis

According to Cole and Maxwell (2003, question 6: What are the effects of ran-

dom measurement error?), unmodeled measurement error variance can cause

both under- and overestimation of other model parameters. A natural extension

of the simple mediation model that we have illustrated involves the use of latent

variables, each of which can be indicated by multiple manifest variables. By

explicitly modeling error variance (�) in the measurement equation, the bias

in parameter estimation can be almost completely resolved, provided that the

model is otherwise correctly specified. Further, psychometric properties of the

measurement instruments (e.g., tau-equivalence) can be evaluated by equating

some loading parameters in the L matrix. As for studying mediation, dynamics

of the process can be examined at the latent level while taking into consideration

the factorial structure of the data.

4.5. Use of Exogenous Variables

Exogenous variables, or fixed external inputs, may enter into the observa-

tion equation, the transition equation, or both (e.g., Lutkepohl, 2005, p. 613;

Shumway & Stoffer, 2011, p. 320). In this case, the linear Gaussian SSM is

extended to

yt ¼ tt þ LtZt þ �txt þ et; et � MVNð0;�tÞZt ¼ at þ BtZt�1 þ�txt þ zt; zt � MVNð0;CtÞ:

Perhaps the most common purpose for including exogenous variables in sta-

tistical models is to account for the variance in some random component (e.g.,

Snijders & Bosker, 2011). Molenaar, de Gooijer, and Schmitz (1992) used a dis-

crete time variable in the transition equation to accommodate a linear time trend.

Beyond parameter estimation in time-varying SSM, Molenaar (1994, 2010a,

2010b) presented the theory, methods, and application of optimal control in psy-

chopathological processes. Basically, a feedback function is derived after the


134

SSM parameters are estimated to determine the optimal level of the external

inputs such that it is possible to manipulate the external inputs (e.g., the insulin

dose) by the controller (e.g., the therapist) to guarantee that the outcome variable

(e.g., the blood glucose level of a patient) will be as close as possible to the

desired level. This type of application presents a grand opportunity to develop

more effective personalized treatment, as opposed to the general dosage of the

clinical medication which is not optimal, less effective, and often brings some

undesired side effects.

Moreover, it is possible to collect data with exogenous variables in the

multiple-subject time-series design. If the exogenous variable is time varying

within a person, the multiple-group analysis discussed before might be applied

(e.g., Molenaar, 2010b). If the exogenous variable is time-invariant within a per-

son but varying between persons (i.e., a Level-2 variable), the multilevel (or ran-

dom coefficient) state space framework can be applied (Song & Ferrer, 2012). In

both scenarios, moderated mediation (or other complex interactions) can be

examined (Card, 2012; Preacher, Rucker, & Hayes, 2007). Future research on

this topic is warranted.

4.6. Missing Data

An attractive feature of SSM is that missing values in the time series data are

easily handled by the KF algorithm. The full-information maximum likelihood

(FIML) procedure varies the dimension of the data vector for observations that

contain one or more missing values. A computationally easier variant of the

FIML procedure is to zero out the missing values and retain the same dimension

of the data vector throughout the observations. No additional effort is needed to

preprocess the missing values as in multiple imputation procedures. Shumway

and Stoffer (2011, chapter 6, subsection 6.4) outline the details of the FIML pro-

cedure and its easier variant, and they also describe the necessary modifications

for the expectation–maximization optimization algorithm.

4.7. Software Implementation

Estimating time-varying SSMs is often a difficult task in that extensive pro-

gramming skills are required to write one’s own software. EKFIS, a Fortran pro-

gram developed by Peter Molenaar to implement the Extended KF with Iteration

and Smoothing algorithm has been used to illustrate the examples in several arti-

cles and chapters (e.g., Molenaar et al., 2009; Molenaar & Ram, 2009, 2010; Sin-

clair & Molenaar, 2008). However, the flexibility of time-varying SSM can

become a liability as well, as EKFIS requires the ability to write and compile For-

tran code (Molenaar & Ram, 2010). Other authors have implemented the

(extended) KF algorithm using MATLAB and Ox/Ssfpack to estimate time-

varying SSMs (e.g., Chow, Hamaker, & Allaire, 2009; Chow, Hamaker, Fujita,

& Boker, 2009; Chow, Zu et al., 2011; Zu, 2008). The programming efforts

Gu et al.

135

involved are still, unfortunately and inevitably, demanding, which is one of the

reasons for the scarcity of modeling work along these lines.

The programming task is relatively easier for time-invariant SSM than for its

time-varying counterpart. In 2011, several articles were published to illustrate dif-

ferent software packages (e.g., EViews, MATLAB, R, SAS, Stata, and several oth-

ers) in a special volume (Vol. 41) of the Journal of Statistical Software. However,

each software package has certain limitations. Readers are urged to consult this

special volume to get a flavor of the package with which they are the most familiar.

It is worth noting that MKFM6, a Fortran program provided by Dolan (2005), and

a SAS/IML program, provided by Gu and Yung (2013), are available to estimate

time-invariant linear Gaussian SSMs. MKFM6 is free and has been used by several

authors (e.g., Chow, Ho, Hamaker, & Dolan, 2010; Hamaker et al., 2005; Zhang

et al., 2008). The programming tasks in this article are implemented in SAS/IML,

which was developed by modifying and extending the SAS/IML code provided

by Gu and Yung. All the programs can be obtained by request from the first author.

Finally, a Bayesian approach to parameter estimation for DFM and SSM is

emerging (e.g., Bhattacharya, Ho, & Purkayastha, 2006; Bhattacharya & Maitra,

2011; Chow, Tang, Yuan, Song, & Zhu, 2011; Song & Ferrer, 2012; Zhang &

Nesselroade, 2007). This newer approach requires estimation methods that are

computationally heavy (e.g., Markov chain Monte Carlo). The rapid develop-

ment of specialized software programs (e.g., WinBUGS, Mplus, OpenBUGS),

however, makes use of Bayesian methods both manageable and appealing.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research,

authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication

of this article.

Notes

1. Application of the state space model (SSM) is not restricted to single-subject time

series data. For example, Chow, Ho, Hamaker, and Dolan (2010) provide an

example in which SSM is applied to cross-sectional data. Moreover, we will

discuss the extension of multiple-subject time series data in Subsection 4.1.

2. Note that our definition of stationarity is consistent with that used in the time-

series literature, but not strictly parallel to the same term described in the med-

iation literature (Cole & Maxwell, 2003; Kenny, 1979).

3. More discussion is provided in Subsection 4.4.

4. Proof of the Kalman filter (KF) algorithm can be found, for instance, in Lutkepohl

(2005, pp. 630–631) or Shumway and Stoffer (2011, pp. 326–327).


136

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Authors

FEI GU is an assistant professor at the Department of Psychology, McGill University,

1205 Doctor Penfield Avenue, Montreal, QC H3A 1B1, Canada; e-mail: fei.gu@mc

gill.ca. He is interested in innovative applications of state space methods. Specifically,

his work extends the state space model to analyze cross-sectional, multilevel, and

(intensive) longitudinal data from multiple subjects and unifies many existing statistical

models into the state space approach.

KRISTOPHER J. PREACHER is an assistant professor at the Psychology & Human

Development,Vanderbilt University, PMB 552, 230 Appleton Place, Nashville, TN

37203-5721; e-mail: [email protected]. His research focuses on the use

of latent variable analysis and multilevel modeling to analyze longitudinal and

correlational data. Other interests include developing techniques to address mediation,

moderation, and model evaluation and selection.

EMILIO FERRER is a professor at the Department of Psychology University of

California, Davis, One Shields Avenue, Davis, CA 95616-8686; e-mail: eferrer@ucda

vis.edu. He is interested in methods to analyze change and intraindividual variability, in

particular latent growth analysis and linear and nonlinear dynamical systems. His

current research in this area involves techniques to model dyadic interactions and devel-

opmental processes underlying fluid reasoning.

Manuscript received July 10, 2013

Revision received November 8, 2013

Accepted January 21, 2014

Gu et al.

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Documents

A State Space Modeling Approach to Mediation Analysis